On the study of solutions for a non linear differential equation on compact Riemannian Manifolds

In this paper we study the existence of solutions for a class of non-linear differential equation on compact Riemannian manifolds. We establish a lower and upper solutions' method to show the existence of a smooth positive solution for the equation (EQ1) \begin{equation} \label{E4} \Delta u \ + \ a(x)u \ = \ f(x)F(u) \ + \ h(x)H(u), (EQ1) \end{equation} where \ $a, \ f, \ h$ \ are positive smooth functions on $M^n$, a $n-$dimensional compact Riemannian manifold, and \ $F, \ H$ \ are non-decreasing smooth functions on $\mathbb{R}$. In \cite{djadli} the equation (EQ1) was studied when $F(u)=u^{2^{\ast}-1}$ and $H(u)=u^q$ in the Riemannian context, i.e., \begin{equation} \label{E3} \Delta u \ + \ a(x)u \ = \ f(x)u^{2^{\ast}-1} \ + \ h(x)u^q, (EQ2) \end{equation} \nd where \ $0 \ < \ q \ < 1$. In \cite{correa} Corr\^ea, Gon\c{c}alves and Melo studied an equation of the type equation (EQ2), in the Euclidean context.